Theorem mulgt0d

Description: The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ltd.2	$$⊢\phi \to B\in ℝ$$
	mulgt0d.3	$$⊢\phi \to 0<A$$
	mulgt0d.4	$$⊢\phi \to 0<B$$
Assertion	mulgt0d	$$⊢\phi \to 0<AB$$

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltd.2	$$⊢\phi \to B\in ℝ$$
3		mulgt0d.3	$$⊢\phi \to 0<A$$
4		mulgt0d.4	$$⊢\phi \to 0<B$$
5		mulgt0	$$⊢\left(\left(A\in ℝ\wedge 0<A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to 0<AB$$
6	1 3 2 4 5	syl22anc	$$⊢\phi \to 0<AB$$